Newton's Law of Gravitation ratio

Homework Statement

A mass M is split into two parts, m and M - m, which are then separated by a certain distance. What ratio m/M maximizes the magnitude of the gravitational force between the parts?

F=Gm1m2/d^2

The Attempt at a Solution

I first just tried plugging in M - m and m in for m1 and m2 to see what I get.

F=(GMm-Gm)/d^2

I believe that I will have to take a derivative to find the maximum, but I am confused as to what I should take the derivative with respect to.

you should not need a derivative. look at extreme cases. if the ratio of M/m was 1000, what would the force be like? what if the ratio was smaller, say 2?

That is true, but I need to prove that is 1/2. The only way I can think of doing that is with a derivative. Although I do see the logic in what you are saying about taking the extreme cases.

try assigning integers to the variables M and d, since they do not change based upon different m values. also, recheck the GMm-Gm part of the equation. I got something else for if i plugged in the values like you did.

Starting with the regular equation for the force of gravity:

$$F=G\frac{m_0m_1}{d^2}$$

Let $$m_0=M-m$$ and $$m_1=m$$

Substitute into the first equation:

$$F=G\frac{(M-m)m}{d^2}=G\frac{Mm-m^2}{d^2}$$

To maximize $$F$$, $$Mm-m^2$$ has to be maximized.

Take its derivative with respect to $$m$$ and set it to zero:

$$M-2m=0$$

Rearrange:

$$M=2m$$

Rearrange:

$$\frac{1}{2}=\frac{m}{M}$$